Method for ascertaining a rough trajectory from a specified contour

ABSTRACT

The invention relates to a method for ascertaining a rough trajectory from a specified contour for controlling a machine tool which has at least two mutually redundant drive devices for carrying out superimposed movements, wherein the contour is determined by a contour function (Pj, pj) which is defined in portions by contour nodal points P0−Pn+1 and respective contour portion functions p0−pn, wherein a respective contour portion function pj connects two adjacent contour nodal points Pj, Pj+1, wherein the rough trajectory is determined by a rough trajectory function (Qj, qj) which is defined in portions by rough trajectory nodal points Q0−Qn+1 and respective rough trajectory portion functions q0−qn, wherein a respective rough trajectory portion function q connects two adjacent rough trajectory nodal points Qj, Qj+1, wherein, for each contour nodal point Pj, a respective assigned rough trajectory nodal point Qj is ascertained in such a manner that a difference in the gradients of the two adjacent rough trajectory portion functions qj−1, qj which contain this rough trajectory nodal point Qj is minimal and that the distance of the contour nodal point Pj from the rough trajectory nodal point Qj satisfies a specified distance condition.

The present invention relates to a method for ascertaining a roughtrajectory from a specified contour for controlling a machine tool whichhas at least two mutually redundant drive devices for carrying outsuperimposed movements.

BACKGROUND OF THE INVENTION

Such machine tools are used for example in milling, laser cutting,water-jet cutting or engraving wood, metal or plastics workpieces or asdrafting machines (plotters) in order to be able to produce workpiecesor drawing lines having a specified two- or three-dimensional contour. Astationary or also a moving, in particular rotating, tool may be movedwith the assistance of the drive devices along the specified contour,such that once machining is complete the workpiece has a desired finalcontour.

Depending on the course of the desired final contour, the tool often hasto cover relatively large distances within a short time and isconsequently also exposed to severe acceleration and/or decelerationforces. Machine tools which have just a single drive device for eachdesired direction of movement of the tool rapidly reach their limits ofperformance in this respect. The speed of machining often has to bereduced to below an acceptable level in order to remain within the speedand/or acceleration limits of the drive device.

This is avoided by using what are known as redundant drive devices foreach direction of movement of the machine tool. To this end, alow-dynamic drive is provided which is capable of moving over relativelylarge displacements but, due to its relatively high mass, has only lowmotion dynamics. In addition, a second, high-dynamic drive is providedwhich, on the one hand, can be displaced by means of the low-dynamicdrive and, on the other hand, is capable is displacing the tool at highspeed and high acceleration or deceleration, wherein however the maximumdisplacement of the high-dynamic drive device is generally limited.

In order to be able to control such a machine tool with redundant drivedevices for a respective direction of movement, it is conventional todivide the contour with which the workpiece is to be machined into arough trajectory and a fine trajectory. The low-dynamic drive is herecontrolled with the rough trajectory data while the high-dynamic driveis simultaneously controlled with the fine trajectory data.

Dividing the contour into a rough trajectory and a fine trajectory andcorresponding control of the machine tool is in principle known and isdescribed for example in DE 10355614 B4 and EP 0594699 B1. Whencalculating the rough trajectory, at least the limited displacement ofthe high-dynamic drive must be taken into account since the workpiecewould otherwise be incorrectly machined. Further limiting parameters areadvantageously also taken into account in calculating the trajectory.This generally results in the rough trajectory comprising somewhatlow-frequency motion components while the fine trajectory hashigh-frequency motion components. In general, the rough trajectory andthe fine trajectory are calculated such that a rough trajectory isascertained and then the fine trajectory is determined by subtractingthe rough trajectory from the contour.

EP 1963935 B1 describes a further method for ascertaining for roughtrajectory which is to be travelled in positionally guided manner. Aninitial trajectory to be travelled is here specified to a computer,wherein the initial trajectory is described by an initial function suchthat a corresponding position on the initial trajectory is in each casedetermined by inserting a scalar trajectory parameter into the initialfunction, wherein the scalar trajectory parameter is other than time andis characteristic of a path travelled along the initial trajectory. Thecomputer subjects the initial trajectory to filtering with a low-passcharacteristic as a function of the scalar trajectory parameter and inthis manner ascertains a rough function, such that a correspondingposition on the rough trajectory is in each case determined by insertingthe scalar trajectory parameter into the rough function. The low-passcharacteristic here relates to the scalar trajectory parameter. Thecomputer ascertains the rough function such that the distance of therough trajectory from the initial trajectory is always below apredetermined bound irrespective of the value of the scalar trajectoryparameter.

In other words, EP 1963935 B1 proposes a method for calculating a roughfunction for travel by a low-dynamic drive which is calculated such thatan initial trajectory dependent on a travel parameter is filtered inrelation to this travel parameter. The low-pass filtered function ischecked as to whether a distance of this function from the initialtrajectory is below a predetermined bound over the entire range of thetravel parameter. On the basis of the low-pass filtered function,further approximations may optionally gradually be carried out in orderto ascertain the rough trajectory providing that the above-stated boundis observed.

The doctoral thesis “Steuerung von Werkzeugmaschinen mit redundantenAchsen” [control of machine tools with redundant axes] by Mr. Marco Bockof the faculty of mathematics and computer science, physics andgeography at Justus Liebig University Gießen submitted in August 2010(http://geb.uni-giessen.de/geb/volltexte/2011/7970/pdf/BockMarco_2010_11_19.pdf)describes various further methods for ascertaining a rough trajectoryfrom a specified contour for controlling a machine tool.

According to a first exemplary embodiment, the rough function may beascertained by initially ascertaining first characteristic intermediatevectors with control points of a spline representation of the initialtrajectory. On this basis, second characteristic intermediate vectorswhich contain control points and define a second intermediate trajectorymay be ascertained from the first characteristic intermediate vectors ofthe spline representation. The control points may be ascertained byweighted or unweighted averaging of pairs of immediately successiveintermediate vectors of the first sequence. On this basis, thirdintermediate vectors may be calculated in corresponding manner. Afterthis double determination of the intermediate trajectory, it must thenbe ascertained whether a geometric distance of the intermediatetrajectory as a rough function from the initial trajectory is below thespecified bound along the trajectory parameter. The spline vectors ofthe initial trajectory may to this end be compared with the splinevectors of the intermediate trajectory of the rough function, whereinthe maximum value of these distances provides an upper distance limitwhich may in turn be compared with the bound for observance of thespecified criterion.

In a second exemplary embodiment, respective trajectory positions on theinitial trajectory may be ascertained for a plurality of scalar valuesof the trajectory parameter on the basis of a spline representation ofthe initial trajectory. On the basis of these pairs of values, a firstintermediate trajectory is defined by the above-stated sampling. Asecond intermediate trajectory of the rough function within the intervalof the scalar trajectory parameter may be determined by weighted orunweighted averaging of the positions on the first intermediatetrajectory. The second intermediate trajectory may be compared withregard to observance of the bound with the initial trajectory or withthe first sampled intermediate trajectory taking account of an auxiliarybound.

Depending on the specified contour, it may happen that known trajectorydivision methods are incapable of supplying satisfactory results. Undercertain circumstances, known methods may be highly computationallyintensive and require a correspondingly long computing time. It is alsoconceivable for the machining time arising from trajectory division notto correspond to the physical capabilities of the machine tool and thusto be extended.

The problem addressed by the invention is therefore that of providing amethod of the initially stated kind which is improved in comparison withknown methods.

SUMMARY OF THE INVENTION

The problem is solved by a method having the features of claim 1.Advantageous developments of the method are indicated in the dependentclaims.

A method is proposed for ascertaining a rough trajectory from aspecified contour for controlling a machine tool which has at least twomutually redundant drive devices for carrying out superimposedmovements, wherein the contour is determined by a contour function(P_(j), p_(j)) which is defined in portions by contour nodal points P₀to P_(n+1) and respective contour portion functions p₀ to p_(n), whereina respective contour portion function p_(j) connects two adjacentcontour nodal points P_(j), P_(j+1), wherein the rough trajectory isdetermined by a rough trajectory function (Q_(j), q_(j)) which isdefined in portions by rough trajectory nodal points Q₀ to Q_(n+1) andrespective rough trajectory portion functions q₀ to q_(n), wherein arespective rough trajectory portion function q_(j) connects two adjacentrough trajectory nodal points Q_(j), Q_(j+1), wherein, for each contournodal point P_(j), a respective assigned rough trajectory nodal pointQ_(j) is ascertained in such a manner that a difference in thegradients, in particular a magnitude of the difference in the gradients,of the two adjacent rough trajectory portion functions q_(j−1), q_(j)which contain this rough trajectory nodal point Q_(j) is minimal andthat the distance of the contour nodal point P_(j) from the roughtrajectory nodal point Q_(j) satisfies a specified distance condition.

The contour nodal points P_(j) are two- or three-dimensional samplingpoints (x_(j) y_(j)) or (x_(j), y_(j), z_(j)) which reproduce thespecified contour for j=0 to n+1. The two-dimensional case, which mayalso be transferred into three dimensions, is considered below.

The proposed method can be carried out with little computational effort.It yields a rough trajectory function which is very smooth and, thanksto the minimised gradient change in the rough trajectory nodal points,allows the low-dynamic drive device to be driven at the highest possiblespeed and in particular with low acceleration or deceleration values andlittle jerkiness. The specified distance condition here provides aconstraint for the optimisation problem to be solved and ensures thatthe distance between the rough trajectory and the original contourremains within specified limits.

In contrast with the prior art, the method according to the inventiondoes not involve low-pass filtering by weighted or unweighted averagingof individual values. In addition, checking as to whether the distanceof the rough trajectory from the contour is always below a predeterminedbound independently of a value of a scalar trajectory parameter is notprovided and is also not necessary since a corresponding check isalready inherently provided by a specified distance condition whichunderlies the mathematical optimisation as a constraint.

It should be noted at this point that the stated contour which providesthe basis for trajectory division according to the invention need notnecessarily be the final contour of the workpiece to be machined. It ishere optionally also possible to take account of material removal, forexample brought about by the tool. For example, account may be taken ofthe diameter of the milling head when a milling cutter is used.

Advantageously, the specified distance condition requires that thedistance between the contour nodal point P_(j) and the assigned roughtrajectory nodal point Q_(j) is less than or equal to a predeterminedlimit value Δ. This constitutes a specific development of the constraintexplained above for solving the optimisation problem.

The predetermined limit value Δ is appropriately based on thedisplacement of the high-dynamic drive device.

According to an advantageous development, the contour function (P_(j),p_(j)) is defined in a plurality of dimensions, wherein the distancecondition, according to which the distance between the contour nodalpoint P_(j) and the assigned rough trajectory nodal point Q_(j) is lessthan or equal to a predetermined limit value Δ, requires that, in eachdimension, the distance be less than or equal to a predetermined limitvalue Δ for this dimension, wherein in particular the limit values Δ areequal for all dimensions. Ascertaining the rough trajectory nodal pointsmay be further simplified as a consequence since each dimension can behandled separately.

According to a further advantageous development, ascertaining therespective assigned rough trajectory nodal point Q_(j) for each contournodal point P_(j) requires that those rough trajectory nodal pointsQ_(j) for which the sum of all the squared differences between thegradients of two in each case adjacent rough trajectory portionfunctions q_(j−1), q_(j) of the rough trajectory function (Q_(j), q_(j))is minimal are ascertained. This optimisation provides particularly goodresults in terms of a maximally smooth gradient profile of the roughtrajectory function (Q_(j), q_(j)). This optimisation problem maystraightforwardly be reformulated as a matrix representation and is thendenoted a quadratic problem or quadratic program. Suitable solutionmethods, in particular numerical solution methods for such anoptimisation problem are familiar to a person skilled in the art andexamples are set out below.

Advantageously, the contour function (P_(j), p_(j)) is defined in aplurality of dimensions, wherein ascertaining those rough trajectorynodal points Q_(j) for which the sum of all the squared differencesbetween the gradients of two in each case adjacent rough trajectoryportion functions q_(j−1), q_(j) of the rough trajectory function Q_(j),q_(j) is minimal requires that the coordinates of the rough trajectorynodal points Q_(j) are separately ascertained in each dimension. Thestated sum to be minimised is accordingly likewise considered separatelyfor each dimension. This consequently further reduces computationaleffort.

According to an advantageous development, the rough trajectory portionfunctions q₀ to q_(n) are formed by respective linear functions. Inother words, the rough trajectory nodal points Q₀ to Q_(n+1) are in eachcase connected by straight lines. As a result, no complex interpolationsteps are required, which means that the computing time required forgenerating the rough trajectory can be distinctly reduced.

Alternatively, it is however also possible to generate the roughtrajectory portion functions q₀ to q_(n) by means of a splineinterpolation of the rough trajectory nodal points Q₀ to Q_(n+1). Astill smoother course of the rough trajectory and thus still less jerkyoperation of the drive devices may be achieved as a consequence.

A rough trajectory may generally be ascertained from the overall contourby one-off application of the method. In many cases, however, the courseof the contour only known in portions and the further course to the endof the contour is still unknown to the method or the number of contourpoints P_(j) is too large for rapid application of the method. It isadvantageously proposed to this end to apply the method iteratively,such that a rough trajectory may be ascertained in portions in relationto the contour. Accordingly, in an advantageous further development ofthe method for ascertaining the rough trajectory, a first roughtrajectory portion of rough trajectory nodal points Q₀ to Q_(n0+1) andrespective rough trajectory portion functions q₀ to q_(n0) may bedetermined from contour points P₀ to P_(n0+1), with n0<n and k=0. Thefurthest distant P_(n0+1) is here obtained from the distance norm∥P_(n0+1)−P₀∥_(∞)>2Δ. Distance Δ may be for example a maximum(unidirectional) displacement of a drive device, preferably of thehigh-dynamic drive device. Thereafter, further rough trajectory portionsof rough trajectory nodal points Q_(k) to Q_(nk+1) and respective roughtrajectory portion functions q_(k) to q_(nk) with nk=k+1, . . . , n andk<n may be determined in subsequent iterations with k>0 with regard tothe distance condition 2Δ between points P_(k) and P_(nk+1), i.e.∥P_(nk+1)−P_(k)∥_(∞)>2Δ. The iterations are carried out until nk=n.

At least the rough trajectory point Q_(k), preferably at least the tworough trajectory points Q_(k−1), Q_(k) of the preceding rough trajectoryportion, may advantageously be set as the starting point for asubsequent iteration, wherein contour points P_(k+1), to P_(nk+1) areotherwise considered.

In this manner, a contour may be converted into a rough trajectory inportions. For a first contour portion P₀ to P_(n0+1), the method isapplied from index point 0 up to an index point no which is defined bythe distance condition that the starting and end points P₀ and P_(n0+1)do not exceed a distance limit 2Δ. For a subsequent rough trajectoryportion, a new starting value k<nk may be used as the starting valueand, on the basis of point P_(k), a point P_(nk) maximally distanttherefrom which satisfies the distance condition 2Δ may in turn bedefined. At least the rough trajectory point Q_(k), in particular thetwo points Q_(k−1), Q_(k) of the preceding rough trajectory portion,is/are used as the starting point for the subsequent iteration. Jumps ontransition from one rough trajectory portion to the next mayconsequently be avoided and the gradient in the transitional zoneminimised.

The preceding method involves building up the rough trajectory inportions. It has proven advantageous for iterative calculation of roughtrajectory portion functions of the further rough trajectory portion(s)of the index k to be shifted by one index value and thus k:=k+1 appliesfor a subsequent iteration. The window of the rough trajectory portionto be calculated is consequently shifted just by the distance of acontour point P_(j). Rough trajectory points Q₀ to Q_(n0+1) are thusinitially calculated, wherein according to the distance condition theindex value n0 with ∥P_(n0+1)−P₀∥_(∞)>2Δ is obtained, i.e. the smallestpossible index value n0 is found, such that the contour point P_(n0+1)is at a distance from the starting point P₀ which is just larger than2Δ. In the next iteration, rough trajectory points Q₁ to Q_(n1) etc. areascertained, wherein the index value n1 is in turn obtained from thedistance condition ∥P_(n1+1)−P₁∥_(∞)>2Δ. Q₀ and Q₁ of the first roughtrajectory portion are here advantageously used as starting values forthe second iteration instead of points P₀ and P₁. Thus, just one or morefurther contour points P_(j) of the original course of the contour arenewly added for the next rough trajectory portion and the first point orpoints Q₀ and Q₁ of the preceding iteration are included. The previouslycalculated rough trajectory points Q_(k−1) and Q_(k) thus define thestarting points of the rough trajectory portion of the subsequentiteration and the method is continued until nk=n. Shifting each furtherrough trajectory portion by 1 has proven to be optimal for thedetermination of the rough trajectory function.

DRAWINGS

Further advantages are revealed by the drawings and the associateddescription of the drawings. The drawings show exemplary embodiments ofthe invention. The drawings, description and claims contain numerousfeatures in combination. A person skilled in the art will expedientlyalso consider these features individually and combine them intomeaningful further combinations.

In the figures:

FIGS. 1-4 show schematic diagrams of a specified contour and anassociated rough trajectory ascertained by means of the method accordingto the invention.

The method according to the invention is described below by way ofexample on the basis of a trajectory division of a two-dimensionalcontour which is defined in an (X, Y) plane, wherein generalisation toother dimensions is, of course, possible. Trajectory division proceedsfor example for a machine tool which has two redundant drive devices foreach direction of movement. The contour may be described, for example,by first, third or fifth order splines, as is conventional for theoperation CNC machine tools. Other contour descriptions may, however,also be available.

The starting point is a contour which is defined at least by contournodal points P_(j)=(x_(j) ⁰, y_(j) ⁰), j=0, . . . , n+1 in a plane (X,Y).

A length s_(j) between two adjacent contour nodal points P_(j), P_(j+1)is defined by:

s _(j) =∥P _(j+1) −P _(j)∥₂=√{square root over ((x _(j+1) ⁰ −x _(j)⁰)²+(y _(j+1) ⁰ −y _(j) ⁰)²)},j=0, . . . ,n

For simplification, it is assumed that two adjacent contour nodal pointsP_(j), P_(j+1) are connected by straight lines with gradients:

${{dx_{j}^{0}} = \frac{x_{j + 1}^{0} - x_{j}^{0}}{s_{j}}},{j = 0},\ldots,n$${{dy_{j}^{0}} = \frac{y_{j + 1}^{0} - y_{j}^{0}}{s_{j}}},{j = 0},\ldots,n$

This contour is now divided by a rough trajectory function (Q_(j),q_(j)) for controlling a low-dynamic drive device, which function isdefined by rough trajectory nodal points Q_(j)=(x_(j), x_(y)), j=0, . .. , n+1 and rough trajectory portion functions connecting the roughtrajectory nodal points Q_(j)□_(□), □=0, . . . , □, having a smoothcourse and therefore a high advance speed with low acceleration anddeceleration values and little jerkiness. Achieving this means that thedifference in gradient of two adjacent rough trajectory portionfunctions q_(j−1), q_(j), or more precisely the absolute value of thedifference in the gradients |dx_(j)−dx_(j−1)| or |dy_(j)−dy_(j−1)|, in arough trajectory nodal point Q_(j) must be small in order to obtain thedesired gentle transitions.

Since a respective drive device is assigned to each dimension x, y, thedimensions x, y may be mutually independently considered. Thecalculation steps are described below solely on the basis of the xcomponent. The y and optionally z components are ascertained incorresponding manner.

The condition, according to which the change in gradient between twoadjacent rough trajectory portion functions q_(j−1), q_(j) should besmall, may be equivalently expressed for example in that a function

${f(x)} = {\frac{1}{2}{\sum\limits_{j = 1}^{n}\left( {{dx}_{j} - {dx_{j - 1}}} \right)^{2}}}$

should be minimised. This corresponds to a least squares method. After anumber of transformations, the function f(x) may be expressed as

f(x)=1/2x ^(T) Qx,

wherein Q is a sparsely populated, symmetrical and positive semidefiniteband matrix which accommodates reciprocal items of length information siand x denotes a vector x=(x₀, . . . , x_(n+1))^(T) of the individualcomponents.

As a constraint for solving this optimisation problem, the roughtrajectory nodal points Q_(j), j=0, . . . , n+1 must be situated withina specified window with edge length Δ around a respective associatedcontour nodal point P_(j). This may be expressed by the condition∥Q_(j)−P_(j)|_(∞)≤Δ or by the conditions:

|x _(j) −x _(j) ⁰ |≤Δ,j=0, . . . ,n+1,

|y _(j) −y _(j) ⁰ |≤Δ,j=0, . . . ,n+1,

wherein Δ as a rule correlates with the displacement limits of thehigh-dynamic fine drive. On travelling along the contour, the drivedevices should start at contour nodal point P₀ and finish at contournodal point P_(n+1). The problem thus consists in locating points xi_(j)j=0, . . . , n+1 which minimise the function f(x) below the statedconstraints. This problem may be written as

f(x)=1/2x ^(T) Qx→min

x ₀ =x ₀ ⁰

x _(j) ⁰ −Δ≤x _(j) ≤x _(j) ⁰+Δ

x _(n+1) =x _(n+1) ⁰

which is known as a quadratic optimisation problem or quadratic program.

Numerous rapid methods for solving this optimisation problem are knownto a person skilled in the art. Numerical methods such as gradientmethods, active set methods, inner point methods or the class of Krylovsubspace methods, in particular conjugate gradient methods, may bementioned by way of example.

Once the rough trajectory nodal points Q₀, Q_(j+1) have been ascertainedin this manner, the rough trajectory portion functions q₀ to q_(j) maybe determined by connecting adjacent rough trajectory nodal pointsQ_(j), Q_(j+1) with straight lines (corresponding to a first orderspline interpolation) or by third or higher order spline interpolation.

FIGS. 1 to 3 in each case show exemplary contour functions (P_(j),p_(j)) with starting points P₀ and end points P_(n+1) together with therough trajectory functions (Q_(j), q_(j)) ascertained by the methodaccording to the invention as well as the associated starting points Q₀and end points Q_(n+1).

FIG. 4 shows an example of a rough trajectory determination similar toFIG. 1. On the basis of the contour functions (P_(j), p_(j)) representedby a continuous line, all the contour functions up to index n may betransformed into a rough trajectory (Q_(j), q_(j)) shown in dashed linesby individual application of the method. This is already depicted inFIG. 1.

On iterative application of the method, the contour function may besubdivided into subportions k to n_(k) with n_(k)=k+1, . . . , n andk<n, wherein the respective highest index value n_(k) satisfies adistance condition 2Δ. The size of the respective rough contour portionmay be selected such it corresponds to an entire movement space of thehigh-dynamic drive. Starting from a starting point P_(k) for definingn_(k) for the set of contour points to be considered of the respectiverough contour portion, the following therefore applies:

∥P _(nk) −P _(k)∥∞>2Δ

For the first iteration, contour points P₀ to P_(n0+1) are considered,i.e. k=0 and the index value n0 which just corresponds to the distancecondition with distance 2Δ from point P₀ to point P_(n0) is sought.Initial rough trajectory points Q₀ to Q_(n0+1) are determined therefromby applying the method.

For each further iteration, k:=k+1 is set and the maximum index n_(k)with nk=k+1, . . . , n is in turn sought, wherein the subsequent roughtrajectory portion is displaced by an index value distance of 1, i.e.k:=k+1 is calculated. Rough trajectory points from 1 to n1 areaccordingly determined for the second iteration and points j to nj forsubsequent iterations j, wherein the rough points Q_(k−1), Q_(k) of thepreceding rough contour portion are in each case used as the startingpoint(s) of each rough contour portion, and otherwise contour pointsP_(k)+1 to P_(k+1) are considered. The first rough point(s) of thepreviously calculated rough contour portion are thus introduced,remaining unchanged, into each subsequent rough contour portion, whereinthe index window in each case shifts by an index value k:=k+1.Ultimately, in each subsequent iteration, since the index value k isincreased by 1, only one further rough trajectory point is added untilnk=n. The method may be applied at most n-times. Because at least thefirst, preferably the first two, rough trajectory points, or a pluralityof first rough trajectory points of the preceding portion is/are used asstarting points for the subsequent portion, it is possible to ensurethat, on transition from one rough contour portion to the next, therough trajectory gradient in the transitional zone is minimised.

The rough trajectory curve obtained from the iterative method with roughtrajectory points Q _(j) is shown in dash-dotted lines and compared withrough trajectory points Q_(j) obtained by just one-off application ofthe method to all points P₀ to P_(n+1), see FIG. 1. It is clear that animproved approximation to the starting contour may be achieved, whereinan increased number of iterations is required on a reduced number ofcontour points.

1.-10. (canceled)
 11. A method for controlling a machine tool having atleast two mutually redundant drive devices for carrying out superimposedmovements following a contour function, the method comprising:determining a specified contour for controlling the machine tool by thecontour function defined in portions by contour nodal points (P₀ toP_(n+1)) and respective contour portion functions (p₀ to p_(n)), whereina respective contour portion function (p_(j)) connects two adjacentcontour nodal points (P_(j), P_(j+1)); determining a rough trajectory bya rough trajectory function (Q_(j), q_(j)) which is defined in portionsby rough trajectory nodal points (Q₀ to Q_(n+1)) and respective roughtrajectory portion functions (q₀ to q_(n)), wherein a respective roughtrajectory portion function (q_(j)) connects two adjacent roughtrajectory nodal points (Q_(j), Q_(j+1)); ascertaining, for each contournodal point (P_(j)), a respective assigned rough trajectory nodal point(Q_(j)) such that a difference in gradients of the two adjacent roughtrajectory portion functions (q_(j−1), q_(j)) which contain this roughtrajectory nodal point (Q_(j)) is minimal and that a distance of thecontour nodal point (P_(j)) from the rough trajectory nodal point(Q_(j)) satisfies a specified distance condition; and directing movementof one of the at least two mutually redundant drive devices based on therespective assigned rough trajectory nodal point (Q_(j)).
 12. The methodof claim 11, wherein the specified distance condition requires that thedistance between the contour nodal point (P_(j)) and the assigned roughtrajectory nodal point (Q_(j)) is less than or equal to a predeterminedlimit value (Δ).
 13. The method of claim 12, wherein the predeterminedlimit value (Δ) corresponds to a respective maximum displacement of oneof the drive devices.
 14. The method of claim 11, wherein the contourfunction (P_(j), p_(j)) is defined in a plurality of dimensions and inthat a distance condition, according to which the distance between thecontour nodal point (P_(j)) and the assigned rough trajectory nodalpoint (Q_(j)) is less than or equal to a predetermined limit value (Δ),requires that in each dimension the distance is less than or equal to apredetermined limit value (Δ).
 15. The method of claim 14, wherein thepredetermined limit value (Δ) of each dimension is equal for alldimensions.
 16. The method of claim 11, wherein ascertaining therespective assigned rough trajectory nodal point (Q_(j)) for eachcontour nodal point (P_(j)) requires that the rough trajectory nodalpoints (Q_(j)) for which a sum of all squared differences between thegradients of two in each case adjacent rough trajectory portionfunctions (q_(j−1), q_(j)) of the rough trajectory function (Q_(j),q_(j)) is minimal are ascertained.
 17. The method of claim 16, whereinthe contour function (P_(j), p_(j)) is defined in a plurality ofdimensions and in that ascertaining the rough trajectory nodal points(Q_(j)) for which the sum of all the squared differences between thegradients of two in each case adjacent rough trajectory portionfunctions (q_(j−1), q_(j)) of the rough trajectory function (Q_(j),q_(j)) is minimal requires that coordinates of the rough trajectorynodal points (Q_(j)) are separately ascertained in each dimension. 18.The method of claim 11, wherein the rough trajectory portion functions(q₀ to q_(n)) are formed by respective linear functions.
 19. The methodof claim 11, wherein the rough trajectory portion functions (q₀ toq_(n)) are generated via a spline interpolation of the rough trajectorynodal points (Q₀ to Q_(n+1)).
 20. The method of claim 11, wherein afirst rough trajectory portion of rough trajectory nodal points (Q₀ toQ_(n0+1)) and respective rough trajectory portion functions (q₀ toq_(n0)) is determined in a first iteration with regard to a distancecondition (2Δ) from the starting point P₀ to P_(n0+1), with n0<n andk=0, and in subsequent iterations with k>0 further rough trajectoryportions of rough trajectory nodal points (Q_(k) to Q_(nk+1)) andrespective rough trajectory portion functions (q_(k) to q_(nk)) withnk=k+1, . . . , n and k<n are determined with regard to the distancecondition (2Δ) between contour nodal points P_(k) to P_(nk+1), untilnk=n, wherein at least the rough trajectory point Q_(k), preferably atleast the two rough trajectory points Q_(k−1), Q_(k), are set as astarting point for a subsequent iteration.
 21. The method of claim 20,wherein, for the iterative calculation of subsequent rough trajectoryportions of the further rough trajectory nodal points (Q_(k) toQ_(nk+1)) and respective rough trajectory portion functions (q_(k) toq_(nk)), an index k is shifted by 1 and thus k:=k+1 applies for asubsequent iteration.
 22. A system for controlling a machine tool, thesystem comprising: the machine tool comprising at least two mutuallyredundant drive devices for carrying out superimposed movementsfollowing a contour curve; and a computer numerical control (CNC) deviceconfigured to: determine a specified contour for controlling the machinetool by the contour function defined in portions by contour nodal points(P₀ to P_(n+1)) and respective contour portion functions (p₀ to p_(n)),wherein a respective contour portion function (p_(j)) connects twoadjacent contour nodal points (P_(j), P_(j+1)), determine a roughtrajectory by a rough trajectory function (Q_(j), q_(j)) which isdefined in portions by rough trajectory nodal points (Q₀ to Q_(n+1)) andrespective rough trajectory portion functions (q₀ to q_(n)), wherein arespective rough trajectory portion function (q_(j)) connects twoadjacent rough trajectory nodal points (Q_(j), Q_(j+1)), ascertain, foreach contour nodal point (P_(j)), a respective assigned rough trajectorynodal point (Q_(j)) such that a difference in gradients of the twoadjacent rough trajectory portion functions (q_(j−1), q_(j)) whichcontain this rough trajectory nodal point (Q_(j)) is minimal and that adistance of the contour nodal point (P_(j)) from the rough trajectorynodal point (Q_(j)) satisfies a specified distance condition, and directmovement of one of the at least two mutually redundant drive devicesbased on the respective assigned rough trajectory nodal point (Q_(j)).23. The system of claim 22, wherein the specified distance conditionrequires that the distance between the contour nodal point (P_(j)) andthe assigned rough trajectory nodal point (Q_(j)) is less than or equalto a predetermined limit value (Δ).
 24. The system of claim 23, whereinthe predetermined limit value (Δ) corresponds to a respective maximumdisplacement of one of the drive devices.
 25. The system of claim 22,wherein the contour function (P_(j), p_(j)) is defined in a plurality ofdimensions and in that a distance condition, according to which thedistance between the contour nodal point (P_(j)) and the assigned roughtrajectory nodal point (Q_(j)) is less than or equal to a predeterminedlimit value (Δ), requires that in each dimension the distance is lessthan or equal to a predetermined limit value (Δ) for the dimension. 26.The system of claim 25, wherein the predetermined limit value (Δ) ofeach dimension is equal for all dimensions.
 27. The system of claim 22,wherein ascertaining the respective assigned rough trajectory nodalpoint (Q_(j)) for each contour nodal point (P_(j)) requires that therough trajectory nodal points (Q_(j)) for which a sum of all squareddifferences between the gradients of two in each case adjacent roughtrajectory portion functions (q_(j−1), q_(j)) of the rough trajectoryfunction (Q_(j), q_(j)) is minimal are ascertained.
 28. The system ofclaim 27, wherein the contour function (P_(j), p_(j)) is defined in aplurality of dimensions and in that ascertaining the rough trajectorynodal points (Q_(j)) for which the sum of all the squared differencesbetween the gradients of two in each case adjacent rough trajectoryportion functions (q_(j−1), q_(j)) of the rough trajectory function(Q_(j), q_(j)) is minimal requires that coordinates of the roughtrajectory nodal points (Q_(j)) are separately ascertained in eachdimension.
 29. The system of claim 22, wherein the rough trajectoryportion functions (q₀ to q_(n)) are formed by respective linearfunctions.
 30. The system of claim 22, wherein the rough trajectoryportion functions (q₀ to q_(n)) are generated via a spline interpolationof the rough trajectory nodal points (Q₀ to Q_(n+1)).
 31. The system ofclaim 22, wherein a first rough trajectory portion of rough trajectorynodal points (Q₀ to Q_(n0+1)) and respective rough trajectory portionfunctions (q₀ to q_(n0)) is determined in a first iteration with regardto a distance condition (2Δ) from the starting point P₀ to P_(n0+1),with n0<n and k=0, and in subsequent iterations with k>0 further roughtrajectory portions of rough trajectory nodal points (Q_(k) to Q_(k+1))and respective rough trajectory portion functions (q_(k) to q_(nk)) withnk=k+1, . . . , n and k<n are determined with regard to the distancecondition (2Δ) between contour nodal points P_(k) to P_(nk+1), untilnk=n, wherein at least the rough trajectory point Q_(k), preferably atleast the two rough trajectory points Q_(k−1), Q_(k), are set as astarting point for a subsequent iteration.
 32. The system of claim 31,wherein, for the iterative calculation of subsequent rough trajectoryportions of the further rough trajectory nodal points (Q_(k) toQ_(nk+1)) and respective rough trajectory portion functions (q_(k) toq_(nk)), an index k is shifted by 1 and thus k:=k+1 applies for asubsequent iteration.